Schrödinger equation is R-separable in toroidal coordinates

2025/11/10 by Matheus E. Pereira, Alexandre G. M. Schmidt, Pereira, Matheus E. +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Fractional Differential Equations Solutions #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph)

paper · pdf · doi:10.48550/arxiv.2511.08646

openalex publication_date 2025/11/10 · openalex created_date 2025/11/14 · openalex updated_date 2026/07/28

Abstract

We present, for the first time, exact solutions for the Schrödinger equation in Moon and Spencer's toroidal coordinates, and in the electromagnetic toroidal--poloidal coordinate systems. Curiously, both systems present a fractional angular momentum, because of the torus's hole. We achieve these novel solutions using the irregular R-separation of variables, an unexplored approach in Physics, which results in a wavefunction with fractional angular momentum eigenvalues. Numerous solutions for the Schrödinger equation in a variety of external potentials are shown, including an external magnetic field. A plane-wave expansion and a Green function are also presented, setting the stage for future progress in this area.

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